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Communications in Mathematical Physics

, Volume 372, Issue 2, pp 599–656 | Cite as

A Unified Approach to the Klein–Gordon Equation on Bianchi Backgrounds

  • Hans RingströmEmail author
Open Access
Article

Abstract

In this paper, we study solutions to the Klein–Gordon equation on Bianchi backgrounds. In particular, we are interested in the asymptotic behaviour of solutions in the direction of silent singularities. The main conclusion is that, for a given solution u to the Klein–Gordon equation, there are smooth functions ui, i = 0,1, on the Lie group under consideration, such that \({u_{\sigma}(\cdot,\sigma)-u_{1}}\) and \({u(\cdot,\sigma)-u_{1} \sigma-u_{0}}\) asymptotically converge to zero in the direction of the singularity (where \({\sigma}\) is a geometrically defined time coordinate such that the singularity corresponds to \({\sigma\rightarrow-\infty}\)). Here ui, i = 0, 1, should be thought of as data on the singularity. Interestingly, it is possible to prove that the asymptotics are of this form for a large class of Bianchi spacetimes. Moreover, the conclusion applies for singularities that are matter dominated; singularities that are vacuum dominated; and even when the asymptotics of the underlying Bianchi spacetime are oscillatory. To summarise, there seems to be a universality as far as the asymptotics in the direction of silent singularities are concerned. In fact, it is tempting to conjecture that as long as the singularity of the underlying Bianchi spacetime is silent, then the asymptotics of solutions are as described above. In order to contrast the above asymptotics with the non-silent setting, we, by appealing to known results, provide a complete asymptotic characterisation of solutions to the Klein–Gordon equation on a flat Kasner background. In that setting, \({u_{\sigma}}\) does, generically, not converge.

Notes

Acknowledgements

The author would like to acknowledge the support of the Göran Gustafsson Foundation for Research in Natural Sciences and Medicine. This research was funded by the Swedish research council, dnr. 2017-03863.

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© The Author(s) 2019

OpenAccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of MathematicsKTH Royal Institute of TechnologyStockholmSweden

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